subspaces

# subspaces - V, ~w W } . We dene V W to be the intersection...

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Here are some problems about subspaces that you need to try. 1. Let S be a non-empty set of vectors. We deﬁne S = { ~x R n : ~x · ~s = 0 for all ~s S } . (a) Show that S is a subspace. Note that S is not assumed to be a subspace. (b) Show that S ( S ) . 2. Suppose that V is a subspace of R n and that dim( V ) = p . Show that dim( V ) = n - p . 3. Suppose that V,W are subspaces of R n . We deﬁne V + W = { ~v + ~w : ~v
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Unformatted text preview: V, ~w W } . We dene V W to be the intersection of V and W , that is, the set of vectors ~x that are in both V and W . (a) We have seen that V + W is a subspace. Show that V W is a subspace of R n . (b) Show that dim( V + W ) = dim( V ) + dim( W )-dim( V W ). (c) Show that V V = { ~ } 1...
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